Step | Hyp | Ref
| Expression |
1 | | i1frn 23981 |
. . . . 5
⊢ (𝐹 ∈ dom ∫1
→ ran 𝐹 ∈
Fin) |
2 | | difss 3999 |
. . . . 5
⊢ (ran
𝐹 ∖ {0}) ⊆ ran
𝐹 |
3 | | ssfi 8533 |
. . . . 5
⊢ ((ran
𝐹 ∈ Fin ∧ (ran
𝐹 ∖ {0}) ⊆ ran
𝐹) → (ran 𝐹 ∖ {0}) ∈
Fin) |
4 | 1, 2, 3 | sylancl 577 |
. . . 4
⊢ (𝐹 ∈ dom ∫1
→ (ran 𝐹 ∖ {0})
∈ Fin) |
5 | 4 | adantr 473 |
. . 3
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → (ran 𝐹 ∖ {0}) ∈ Fin) |
6 | | i1ff 23980 |
. . . . . . . 8
⊢ (𝐹 ∈ dom ∫1
→ 𝐹:ℝ⟶ℝ) |
7 | 6 | adantr 473 |
. . . . . . 7
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → 𝐹:ℝ⟶ℝ) |
8 | 7 | frnd 6351 |
. . . . . 6
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → ran 𝐹 ⊆ ℝ) |
9 | 8 | ssdifssd 4010 |
. . . . 5
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → (ran 𝐹 ∖ {0}) ⊆
ℝ) |
10 | 9 | sselda 3859 |
. . . 4
⊢ (((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → 𝑥 ∈ ℝ) |
11 | | i1fima2sn 23984 |
. . . . 5
⊢ ((𝐹 ∈ dom ∫1
∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) →
(vol‘(◡𝐹 “ {𝑥})) ∈ ℝ) |
12 | 11 | adantlr 702 |
. . . 4
⊢ (((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (vol‘(◡𝐹 “ {𝑥})) ∈ ℝ) |
13 | 10, 12 | remulcld 10470 |
. . 3
⊢ (((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (𝑥 · (vol‘(◡𝐹 “ {𝑥}))) ∈ ℝ) |
14 | | eldifi 3994 |
. . . . 5
⊢ (𝑥 ∈ (ran 𝐹 ∖ {0}) → 𝑥 ∈ ran 𝐹) |
15 | | 0cn 10431 |
. . . . . . . . . . . 12
⊢ 0 ∈
ℂ |
16 | | fnconstg 6396 |
. . . . . . . . . . . 12
⊢ (0 ∈
ℂ → (ℂ × {0}) Fn ℂ) |
17 | 15, 16 | ax-mp 5 |
. . . . . . . . . . 11
⊢ (ℂ
× {0}) Fn ℂ |
18 | | df-0p 23974 |
. . . . . . . . . . . 12
⊢
0𝑝 = (ℂ × {0}) |
19 | 18 | fneq1i 6283 |
. . . . . . . . . . 11
⊢
(0𝑝 Fn ℂ ↔ (ℂ × {0}) Fn
ℂ) |
20 | 17, 19 | mpbir 223 |
. . . . . . . . . 10
⊢
0𝑝 Fn ℂ |
21 | 20 | a1i 11 |
. . . . . . . . 9
⊢ (𝐹 ∈ dom ∫1
→ 0𝑝 Fn ℂ) |
22 | 6 | ffnd 6345 |
. . . . . . . . 9
⊢ (𝐹 ∈ dom ∫1
→ 𝐹 Fn
ℝ) |
23 | | cnex 10416 |
. . . . . . . . . 10
⊢ ℂ
∈ V |
24 | 23 | a1i 11 |
. . . . . . . . 9
⊢ (𝐹 ∈ dom ∫1
→ ℂ ∈ V) |
25 | | reex 10426 |
. . . . . . . . . 10
⊢ ℝ
∈ V |
26 | 25 | a1i 11 |
. . . . . . . . 9
⊢ (𝐹 ∈ dom ∫1
→ ℝ ∈ V) |
27 | | ax-resscn 10392 |
. . . . . . . . . 10
⊢ ℝ
⊆ ℂ |
28 | | sseqin2 4080 |
. . . . . . . . . 10
⊢ (ℝ
⊆ ℂ ↔ (ℂ ∩ ℝ) = ℝ) |
29 | 27, 28 | mpbi 222 |
. . . . . . . . 9
⊢ (ℂ
∩ ℝ) = ℝ |
30 | | 0pval 23975 |
. . . . . . . . . 10
⊢ (𝑦 ∈ ℂ →
(0𝑝‘𝑦) = 0) |
31 | 30 | adantl 474 |
. . . . . . . . 9
⊢ ((𝐹 ∈ dom ∫1
∧ 𝑦 ∈ ℂ)
→ (0𝑝‘𝑦) = 0) |
32 | | eqidd 2780 |
. . . . . . . . 9
⊢ ((𝐹 ∈ dom ∫1
∧ 𝑦 ∈ ℝ)
→ (𝐹‘𝑦) = (𝐹‘𝑦)) |
33 | 21, 22, 24, 26, 29, 31, 32 | ofrfval 7235 |
. . . . . . . 8
⊢ (𝐹 ∈ dom ∫1
→ (0𝑝 ∘𝑟 ≤ 𝐹 ↔ ∀𝑦 ∈ ℝ 0 ≤ (𝐹‘𝑦))) |
34 | 33 | biimpa 469 |
. . . . . . 7
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → ∀𝑦 ∈ ℝ 0 ≤ (𝐹‘𝑦)) |
35 | 22 | adantr 473 |
. . . . . . . 8
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → 𝐹 Fn ℝ) |
36 | | breq2 4933 |
. . . . . . . . 9
⊢ (𝑥 = (𝐹‘𝑦) → (0 ≤ 𝑥 ↔ 0 ≤ (𝐹‘𝑦))) |
37 | 36 | ralrn 6679 |
. . . . . . . 8
⊢ (𝐹 Fn ℝ →
(∀𝑥 ∈ ran 𝐹0 ≤ 𝑥 ↔ ∀𝑦 ∈ ℝ 0 ≤ (𝐹‘𝑦))) |
38 | 35, 37 | syl 17 |
. . . . . . 7
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → (∀𝑥 ∈ ran 𝐹0 ≤ 𝑥 ↔ ∀𝑦 ∈ ℝ 0 ≤ (𝐹‘𝑦))) |
39 | 34, 38 | mpbird 249 |
. . . . . 6
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → ∀𝑥 ∈ ran 𝐹0 ≤ 𝑥) |
40 | 39 | r19.21bi 3159 |
. . . . 5
⊢ (((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ ran 𝐹) → 0 ≤ 𝑥) |
41 | 14, 40 | sylan2 583 |
. . . 4
⊢ (((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → 0 ≤ 𝑥) |
42 | | i1fima 23982 |
. . . . . 6
⊢ (𝐹 ∈ dom ∫1
→ (◡𝐹 “ {𝑥}) ∈ dom vol) |
43 | 42 | ad2antrr 713 |
. . . . 5
⊢ (((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → (◡𝐹 “ {𝑥}) ∈ dom vol) |
44 | | mblss 23835 |
. . . . . . 7
⊢ ((◡𝐹 “ {𝑥}) ∈ dom vol → (◡𝐹 “ {𝑥}) ⊆ ℝ) |
45 | | ovolge0 23785 |
. . . . . . 7
⊢ ((◡𝐹 “ {𝑥}) ⊆ ℝ → 0 ≤
(vol*‘(◡𝐹 “ {𝑥}))) |
46 | 44, 45 | syl 17 |
. . . . . 6
⊢ ((◡𝐹 “ {𝑥}) ∈ dom vol → 0 ≤
(vol*‘(◡𝐹 “ {𝑥}))) |
47 | | mblvol 23834 |
. . . . . 6
⊢ ((◡𝐹 “ {𝑥}) ∈ dom vol → (vol‘(◡𝐹 “ {𝑥})) = (vol*‘(◡𝐹 “ {𝑥}))) |
48 | 46, 47 | breqtrrd 4957 |
. . . . 5
⊢ ((◡𝐹 “ {𝑥}) ∈ dom vol → 0 ≤
(vol‘(◡𝐹 “ {𝑥}))) |
49 | 43, 48 | syl 17 |
. . . 4
⊢ (((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → 0 ≤
(vol‘(◡𝐹 “ {𝑥}))) |
50 | 10, 12, 41, 49 | mulge0d 11018 |
. . 3
⊢ (((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) ∧ 𝑥 ∈ (ran 𝐹 ∖ {0})) → 0 ≤ (𝑥 · (vol‘(◡𝐹 “ {𝑥})))) |
51 | 5, 13, 50 | fsumge0 15010 |
. 2
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → 0 ≤ Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥})))) |
52 | | itg1val 23987 |
. . 3
⊢ (𝐹 ∈ dom ∫1
→ (∫1‘𝐹) = Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥})))) |
53 | 52 | adantr 473 |
. 2
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → (∫1‘𝐹) = Σ𝑥 ∈ (ran 𝐹 ∖ {0})(𝑥 · (vol‘(◡𝐹 “ {𝑥})))) |
54 | 51, 53 | breqtrrd 4957 |
1
⊢ ((𝐹 ∈ dom ∫1
∧ 0𝑝 ∘𝑟 ≤ 𝐹) → 0 ≤
(∫1‘𝐹)) |